| Let r(n) denote the number of representations of the integer n as a sum of two perfect squares. We begin by determining the average behavior of r(n) when n is confined to a fixed residue class a modulo q, developing an asymptotic formula for the appropriate summatory function of r(n). After this, we investigate the “average” size of the error term in the given asymptotic formula, for all q up to a given size Q less than x.; This type of problem was first studied in the 1960's with Λ( n), the classical von Mangoldt function, in place of r( n). Within the last decade, both Hooley and Vaughan have obtained some general results along these lines. The methods of this thesis are by no means new—the main tools are the Large Sieve Inequality and a transformation of Hooley. For specific behavior of the sequence r( n), we have used results of Vaughan which arise in the treatment of Waring's Problem.; We remark that the sequence r(n) falls outside of the realm of any currently-known results, and may give some indications as to what can be done in greater generality. Some preliminary calculations suggest that the methods used here should enjoy a wider range of applicability, assuming we can determine appropriate conditions on the sequences under consideration. |
